Replacing powers with ReplaceRepeated

Question

I have question about ReplaceRepeated. Consider

$\qquad\mathsf{FullSimplify}\left[\left\{\frac{q^5-q^4}{q^5-2 q^4+2 q^2-q}\right\}\text{//.}\, \left\{q^3\to 1\right\}\right]$

Evaluating the above expression gives

$\qquad\left\{\frac{q^3}{(q-1)^2 (q+1)}\right\}$

but the answer I want is 1/3.

How do I the answer in the form I want?

Answer 1

Try this

Simplify[(q^5-q^4)/(q^5-2q^4+2q^2-q)/.q^n_ -> q^Mod[n,3]]

which returns 1/3 instantly.

That is implicitly replacing each q^3 by 1.